Generalized wavelet transforms and their applications

Leon H. Sibul, Lora G. Weiss, Michael J. Roan

Research output: Contribution to journalConference articlepeer-review

Abstract

Continuous wavelet transforms and linear time-frequency transforms are coefficients of continuous unitary group representations of the affine and Heisenberg groups. Many properties of these transforms that are important for wideband radar and sonar signal processing follow directly from group representation theory. These properties include volume invariance and variance of narrowband and wideband ambiguity functions and wavelet transform domain implementations of detectors and signal estimators. For several radar, sonar, and array processing applications, the basic definition of wavelet and timefrequency representations must be generalized by using unitary representations of other groups and using reproducing kernel Hilbert space (RKHS) inner products in the definition of the linear transforms. The general definition then leads to weighted continuous wavelet transforms where the RKHS is determined by a nonstationary covariance function; generalized wideband ambiguity functions also follow from this general definition along with other important generalizations that arise in wideband array processing and model based signal processing in complex scattering and propagation media. This paper presents the generalized wavelet transform along with the weighted wavelet transforms. The classical narrowband and wideband ambiguity functions are then special cases. The application of generalized transforms to multidimensional transforms for space-time processing are also presented, along with the application to conformal groups for detections and estimation of accelerating scatterers.

Original languageEnglish (US)
Pages (from-to)502-509
Number of pages8
JournalProceedings of SPIE - The International Society for Optical Engineering
Volume3391
DOIs
StatePublished - Mar 26 1998
EventWavelet Applications V 1998 - Orlando, United States
Duration: Apr 13 1998Apr 17 1998

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics
  • Computer Science Applications
  • Applied Mathematics
  • Electrical and Electronic Engineering

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